Average Error: 33.3 → 10.5
Time: 22.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.153478880637207 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.153478880637207 \cdot 10^{+108}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} - b}{2}}}\\

\mathbf{else}:\\
\;\;\;\;-\frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r1378529 = b;
        double r1378530 = -r1378529;
        double r1378531 = r1378529 * r1378529;
        double r1378532 = 4.0;
        double r1378533 = a;
        double r1378534 = r1378532 * r1378533;
        double r1378535 = c;
        double r1378536 = r1378534 * r1378535;
        double r1378537 = r1378531 - r1378536;
        double r1378538 = sqrt(r1378537);
        double r1378539 = r1378530 + r1378538;
        double r1378540 = 2.0;
        double r1378541 = r1378540 * r1378533;
        double r1378542 = r1378539 / r1378541;
        return r1378542;
}
↓
double f(double a, double b, double c) {
        double r1378543 = b;
        double r1378544 = -1.153478880637207e+108;
        bool r1378545 = r1378543 <= r1378544;
        double r1378546 = c;
        double r1378547 = r1378546 / r1378543;
        double r1378548 = a;
        double r1378549 = r1378543 / r1378548;
        double r1378550 = r1378547 - r1378549;
        double r1378551 = 1.8378252714625124e-19;
        bool r1378552 = r1378543 <= r1378551;
        double r1378553 = 1.0;
        double r1378554 = -4.0;
        double r1378555 = r1378548 * r1378554;
        double r1378556 = r1378555 * r1378546;
        double r1378557 = fma(r1378543, r1378543, r1378556);
        double r1378558 = sqrt(r1378557);
        double r1378559 = r1378558 - r1378543;
        double r1378560 = 2.0;
        double r1378561 = r1378559 / r1378560;
        double r1378562 = r1378548 / r1378561;
        double r1378563 = r1378553 / r1378562;
        double r1378564 = -r1378547;
        double r1378565 = r1378552 ? r1378563 : r1378564;
        double r1378566 = r1378545 ? r1378550 : r1378565;
        return r1378566;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.3
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.153478880637207e+108

    1. Initial program 46.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified46.3

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b}{2}}{a}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity46.3

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b}{2}}{\color{blue}{1 \cdot a}}\]

    5. Applied div-inv46.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b\right) \cdot \frac{1}{2}}}{1 \cdot a}\]

    6. Applied times-frac46.4

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b}{1} \cdot \frac{\frac{1}{2}}{a}}\]

    7. Simplified46.4

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b\right)} \cdot \frac{\frac{1}{2}}{a}\]

    8. Simplified46.4

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\]

    9. Taylor expanded around -inf 3.2

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]

    if -1.153478880637207e+108 < b < 1.8378252714625124e-19

    1. Initial program 14.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified14.8

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b}{2}}{a}}\]
    3. Using strategy rm

    4. Applied clear-num15.0

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b}{2}}}}\]

    if 1.8378252714625124e-19 < b

    1. Initial program 54.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified54.4

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b}{2}}{a}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity54.4

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b}{2}}{\color{blue}{1 \cdot a}}\]

    5. Applied div-inv54.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b\right) \cdot \frac{1}{2}}}{1 \cdot a}\]

    6. Applied times-frac54.4

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b}{1} \cdot \frac{\frac{1}{2}}{a}}\]

    7. Simplified54.4

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b\right)} \cdot \frac{\frac{1}{2}}{a}\]

    8. Simplified54.4

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4 \cdot a\right)\right)\right)} - b\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\]

    9. Taylor expanded around inf 7.0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    10. Simplified7.0

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.153478880637207 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))